92 votes
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What, in simplest terms, is gauge invariance?

The reason that it's so hard to understand what physicists mean when they talk about "gauge freedom" is that there are at least four inequivalent definitions that I've seen used: Definition 1: A ...
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38 votes

What role does "spontaneous symmetry breaking" play in the "Higgs Mechanism"?

It is frequently stated the Higgs mechanism involves spontaneous breaking of the gauge symmetry. This is, however, entirely wrong. In fact, gauge symmetries cannot be spontaneously broken. A ...
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33 votes
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If Energy can be converted into mass, why can it not be converted into charge?

You're making some category errors in the question. Energy can't be converted into mass, mass is a form that energy can take. In other words, when energy is "converted" into mass it never stops being ...
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25 votes
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Physical difference between gauge symmetries and global symmetries

The first answer to such a question must always be: A gauge symmetry has no "physical" meaning, it is an artifact of our choice for the coordinates/fields with which we describe the system (cf. Gauge ...
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23 votes

What role does "spontaneous symmetry breaking" play in the "Higgs Mechanism"?

In short: The spontaneous breaking of global U(1) symmetry, rather than local 'gauge symmetry', gives rise to the non-zero vacuum expectation value of Higgs field. This non-zero VEV is an essential ...
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22 votes

What, in simplest terms, is gauge invariance?

I only understood this after taking a class in general relativity (GR), differential geometry and quantum field theory (QFT). The essence is just a change of coordinates systems that needs to be ...
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Covariance in gauge theories: why should the Lagrangian be gauge invariant

We do not start from the assumption that the Lagrangian "should" be invariant under gauge transformations. This assumption is often made because global symmetries are seen as more natural than local ...
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20 votes
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How do symmetries “define” physical laws?

A theory is typically described by a Lagrangian, and varying this gives us the equations of motion of the system. The symmetries you describe are symmetries of the Lagrangian i.e. they are ...
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19 votes

What is $U(1)$ symmetry?

Let us first refer to symmetry generically. When we say a theory is symmetric under $G$ ($G$ some group) we mean that the elements of $G$ transform the states, and the operators of a theory, (in the ...
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18 votes

What, in simplest terms, is gauge invariance?

Since you mentioned coming from a mathematics background, you might find it nice to take an answer in terms of equivalence classes. A gauge theory is physical theory where the observable quantities, ...
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17 votes

What defines a large gauge transformation, really?

Bundles and compactified spacetime A gauge theory cannot be looked at purely locally, it has inherently global features one cannot see locally. The proper mathematical formalization of a Yang-Mills ...
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15 votes
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Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

Let there be given a 4-dimensional real manifold$^1$ $M$. As OP says, the set ${\rm Diff}(M)$ is the group of globally defined $C^{\infty}$-diffeomorphisms $f:M\to M$. The set ${\rm Diff}(M)$ is an ...
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Does classical electrodynamics have $U(1)$ symmetry? If yes, how?

A free "$\mathrm{U}(1)$" gauge theory can never tell whether the gauge group is $\mathrm{U}(1)$ or $\mathbb{R}$ because the only field in the theory, the gauge potential $A$, transforms as $$ A\mapsto ...
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Why can't a real scalar couple to the electromagnetic field?

The fact that the theory is not gauge invariant implies that all degrees of freedom of $A_\mu$ must have physical meaning: This is not the theory of photons where only transverse degrees of freedom ...
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Can we use the term "U(1) gauge invariance" for the free electromagnetic field?

OP has a point. The field $A^\mu$ is a connection, and therefore it lives in the algebra of the gauge group, not in the group itself. In this case, $\mathfrak u(1)=\mathbb R$. At first sight, this is ...
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Examples of "gauging a global symmetry"

Here is a simple example, one of the first you should try to understand. The theory has a free $U(1)$ scalar field $\phi$ in $d+1$ spacetime dimensions, discussed in the modern notation of ...
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When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

I) In general, it is true that if we plug a local Lagrangian $$\tag{1} L\quad \longrightarrow \quad \tilde{L}~=~L+\frac{df}{dt}$$ modified with a total derivative term into the Euler-Lagrange ...
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What are global and local gauge invariance defined as they are?

Multiplying by $e^{i\theta}$ is a rotation of $\theta$ in the complex plane. Physically it changes the phase of a plane wave by an angle $\theta$. This is a global symmetry because we arbitrarily ...
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12 votes

What, in simplest terms, is gauge invariance?

Gauge invariance is simply a redundancy in the description of a physical system. I.e. we can choose from an infinite number of vector potentials in E&M. For example, an infinite number of vector ...
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Why do (can) we impose local gauge invariance?

I'm with you. I don't want to be unprofessional, but I find the whole "breaking causality" thing to be complete bogus. I see absolutely no way that the humble Klein Gordon field "breaks ...
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11 votes
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Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

Comment to the question (v1): It seems OP is conflating, on one hand, a gauge transformation $$ \tilde{A}_{\mu} ~=~ A_{\mu} +d_{\mu}\Lambda $$ with, on the other hand, a gauge-fixing condition, i.e....
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11 votes

What, in simplest terms, is gauge invariance?

These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge ...
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Gauge invariance is just a redundancy. Why is massive abelian gauge field renormalizable but massive non-abelian gauge field nonrenormalizable?

Certainly I admit that power counting law is violated, but why does violation of power-counting have relation with renormalizablity? As per the Dyson-Weinberg power-counting theorem (see Ref.1, ...
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11 votes

Is the Lagrangian in the Standard Model exact or approximate?

You seem to be mixing up a few different things. The Standard Model does not say the up and down quark must have the same mass. The up and down quark form a doublet of the flavor symmetry $SU(2)_F$. ...
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11 votes
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Proving Gauge invariance of Schrodinger Equation

Actually, Schroedinger equation $$ -i\hbar \partial_t \psi+ \big[ -\frac{1}{2m}\big(\frac{\hbar}{i}\nabla-q\vec{A}\big)^2+qV \big]\psi=0\tag{0}$$ under the gauge transformations $$ \psi \rightarrow \...
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10 votes

What, in simplest terms, is gauge invariance?

Here's the most elementary example of a gauge symmetry I can think of. Suppose you want to discuss some ants walking around on a Möbius band. To describe the positions of the ants, it's convenient to ...
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10 votes

Uniqueness of Yang-Mills theory

If you don't impose power-counting renormalizability, there are a host of other possibilities, since higher order derivatives or higher order interactions can be introduced. For example, terms $(Tr(F^...
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10 votes
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Does a gauge transformation leave physics invariant?

The statement because the gauge transformation is not supposed to change anything, it means that every expectation can be calculated equivalently using $ \psi' $ or $ \psi $ isn't particularly ...
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How is the Chern-Simons action well-defined?

The expression of the Chern-Simons functional as an integral over a 3-apace is just a shorthand notation. The integration in the Chern-Simons functional differs from the integration of differential ...
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9 votes
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"Large" gauge transformation doesn't act as do-nothing transformation in QFT: looking for classical analog

That large gauge transformations are not true gauge transformations (i.e. yield physically distinct states) is a purely quantum phenomenon due to a choice of quantization procedure that is present in ...
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